HMM : Viterbi algorithm - a toy example

Transcription

1 MM : Viterbi algorithm - a toy example et's consider the following simple MM. This model is composed of 2 states, (high C content) and (low C content). We can for example consider that state characterizes coding DN while characterizes non-coding DN. The model can then be used to predict the region of coding DN from a given sequence. Sources: For the theory, see Durbin et al (1998); For the example, see Borodovsky & Ekisheva (26), pp 8-81

2 MM : Viterbi algorithm - a toy example Consider the sequence S= CCT There are several paths through the hidden states ( and ) that lead to the given sequence S. Example: P = The probability of the MM to produce sequence S through the path P is: p = p () * p () * p * p () * p * p (C) *... =.5 *.2 *.6 *.2 *.4 *.3 *... =...

3 MM : Viterbi algorithm - a toy example CCT There are several paths through the hidden states ( and ) that lead to the given sequence, but they do not have the same probability. The Viterbi algorithm is a dynamical programming algorithm that allows us to compute the most probable path. Its principle is similar to the DP programs used to align 2 sequences (i.e. Needleman-Wunsch) Source: Borodovsky & Ekisheva, 26

4 MM : Viterbi algorithm - a toy example Viterbi algorithm: principle C C T The probability of the most probable path ending in state k with observation "i" is p l (i, x) = e l ( i)max k ( ) p k ( j, x "1).p kl! probability to observe element i in state l probability of the most probable path ending at position x-1 in state k with element j probability of the transition from state l to state k

5 MM : Viterbi algorithm - a toy example Viterbi algorithm: principle C C T The probability of the most probable path ending in state k with observation "i" is In our example, the probability of the most probable path ending in state with observation "" at the 4th position is:! p l (i, x) = e l ( i)max ( ) p (,4) = e ()max p (C,3) p, p (C,3) p We can thus compute recursively (from the first to the last element of our sequence) the probability of the most probable path. k ( ) p k ( j, x "1).p kl

6 MM : Viterbi algorithm - a toy example C C T T Remark: for the calculations, it is convenient to use the log of the probabilities (rather than the probabilities themselves). Indeed, this allows us to compute sums instead of products, which is more efficient and accurate. We used here log 2 (p).

7 MM : Viterbi algorithm - a toy example C T C T CCT Probability (in log 2 ) that at the first position was emitted by state Probability (in log 2 ) that at the first position was emitted by state p (,1) = = p (,1) = =

8 MM : Viterbi algorithm - a toy example C T C T CCT Probability (in log 2 ) that at the 2nd position was emitted by state Probability (in log 2 ) that at the 2nd position was emitted by state p (,2) = max (p (,1)+p, p (,1)+p ) = max ( , ) = (obtained from p (,1)) p (,2) = max (p (,1)+p, p (,1)+p ) = max ( , ) = (obtained from p (,1))

9 MM : Viterbi algorithm - a toy example C T C T CCT C C T We then compute iteratively the probabilities p (i,x) and p (i,x) that nucleotide i at position x was emitted by state or, respectively. The highest probability obtained for the nucleotide at the last position is the probability of the most probable path. This path can be retrieved by back-tracking.

10 MM : Viterbi algorithm - a toy example C T C T CCT back-tracking (= finding the path which corresponds to the highest probability, ) C C T The most probable path is: Its probability is = 4.25E-8 (remember that we used log 2 (p))

11 MM : Forward algorithm - a toy example Consider now the sequence S= C What is the probability P(S) that this sequence S was generated by the MM model? This probability P(S) is given by the sum of the probabilities p i (S) of each possible path that produces this sequence. The probability P(S) can be computed by dynamical programming using either the so-called Forward or the Backward algorithm.

12 MM : Forward algorithm - a toy example Forward algorithm Consider now the sequence S= C C.5*.3=.15.5*.2=.1

13 MM : Forward algorithm - a toy example Consider now the sequence S= C C.5*.3=.15.15*.5*.3 +.1*.4*.3=.345.5*.2=.1

14 MM : Forward algorithm - a toy example Consider now the sequence S= C C.5*.3=.15.15*.5*.3 +.1*.4*.3=.345.5*.2=.1.1*.6* *.5*.2=.27

15 MM : Forward algorithm - a toy example Consider now the sequence S= C C.5*.3=.15.15*.5*.3 +.1*.4*.3= *.2=.1.1*.6* *.5*.2=

16 MM : Forward algorithm - a toy example Consider now the sequence S= C C.5*.3=.15.15*.5*.3 +.1*.4*.3= *.2=.1.1*.6* *.5*.2= => The probability that the sequence S was generated by the MM model is thus P(S)= Σ =.38432

17 MM : Forward algorithm - a toy example The probability that sequence S="C" was generated by the MM model is P MM (S) = To assess the significance of this value, we have to compare it to the probability that sequence S was generated by the background model (i.e. by chance). Ex: If all nucleotides have the same probability, p bg =.25; the probability to observe S by chance is: P bg (S) = p bg 4 =.25 4 =.396. Thus, for this particular example, it is likely that the sequence S does not match the MM model (P bg > P MM ). NB: Note that this toy model is very simple and does not reflect any biological motif. If fact both states and are characterized by probabilities close to the background probabilities, which makes the model not realistic and not suitable to detect specific motifs.

18 MM : Summary Summary The Viterbi algorithm is used to compute the most probable path (as well as its probability). It requires knowledge of the parameters of the MM model and a particular output sequence and it finds the state sequence that is most likely to have generated that output sequence. It works by finding a maximum over all possible state sequences. In sequence analysis, this method can be used for example to predict coding vs non-coding sequences. In fact there are often many state sequences that can produce the same particular output sequence, but with different probabilities. It is possible to calculate the probability for the MM model to generate that output sequence by doing the summation over all possible state sequences. This also can be done efficiently using the Forward algorithm (or the Backward algorithm), which is also a dynamical programming algorithm. In sequence analysis, this method can be used for example to predict the probability that a particular DN region match the MM motif (i.e. was emitted by the MM model). MM motif can represent a TF binding site for ex.

19 MM : Summary Remarks To create a MM model (i.e. find the most likely set of state transition and output probabilities of each state), we need a set of (training) sequences, that does not need to be aligned. No tractable algorithm is known for solving this problem exactly, but a local maximum likelihood can be derived efficiently using the Baum-Welch algorithm or the Baldi-Chauvin algorithm. The Baum-Welch algorithm is an example of a forward-backward algorithm, and is a special case of the Expectation-maximization algorithm. For more details: see Durbin et al (1998) MMER The UMMER3 package contains a set of programs (developed by S. Eddy) to build MM models (from a set of aligned sequences) and to use MM models (to align sequences or to find sequences in databases). These programs are available at the Mobyle plateform (